July 2016 - mathfun@mailman.xmission.com

[math-fun] Happy Bday, Luca; need 4 billion candles...
by Henry Baker 27 Jul '16

27 Jul '16

FYI -- Possible evidence that life may not have started on Earth, after all... http://www.nytimes.com/2016/07/26/science/last-universal-ancestor.html Meet Luca, the Ancestor of All Living Things "This venerable ancestor was a single-cell, bacterium-like organism. But it has a grand name, or at least an acronym. It is known as Luca, the Last Universal Common Ancestor, and is estimated to have lived some four billion years ago, when Earth was a mere 560 million years old." "Life could have originated anywhere and later been confined to a deep sea environment because of some catastrophic event like the Late Heavy Bombardment, which occurred 4 billion to 3.8 billion years ago. This was a rain of meteorites that crashed into Earth with such force that the oceans were boiled off into an incandescent mist. ... Life is so complex it seems to need many millions of years to evolve. Yet evidence for the earliest life dates to 3.8 billion years ago, as if it emerged almost the minute the bombardment ceased." (Perhaps Luca arrived *with* the Late Heavy Bombardment?)

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[math-fun] Robert Fano, 98, Dies -- Helped Develop Interactive Computers
by Henry Baker 27 Jul '16

27 Jul '16

FYI -- Bob Fano was my undergraduate advisor; his lovely daughter, Paola, worked at BBN for a while. It's bizarre that the NYTimes makes such a big deal out of CTSS; I consider Fano's work in information & coding theory (& whatever he did during WWII, much of which may still be classified) his major accomplishments. He was also an incredibly good teacher, and a wonderful advisor: he finally convinced me that Kronecker may have been somewhat short-sighted ("God created the integers..."), and got me to take probability. http://www.nytimes.com/2016/07/27/technology/robert-fano-98-dies-engineer-w… Robert Fano, 98, Dies; Engineer Who Helped Develop Interactive Computers

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[math-fun] Interactive virtual planimeter
by James Propp 26 Jul '16

26 Jul '16

Does anyone know of a good interactive virtual planimeter that will tell you the signed area of a region that you've specified by dragging around a cursor? Even a version that only works for polygons (where the user clicks successive vertices) would be better than nothing. I've looked on the web, but I couldn't find what I need. Maybe one of you is inclined to build one? I would feature it (with credit, of course) in my upcoming Mathematical Enchantments essay about signed area (later this year). I want something that runs from a browser, so readers of my blog can just click on a link and a new tab will open. I am not interested in explaining the mechanics of how physical planimeters work (though I suppose I wouldn't be opposed to a webpage that explained that too, as long as the visuals associated with the wheels and arms didn't submerge the main message). I'd much rather have extra visuals that reinforce the concept of signed area. For instance, if the demo does just polygons, then every time a new vertex gets added, one could see a new triangle joining the new vertex, the previous vertex, and the origin, with shading or color to indicate whether it's oriented clockwise or counterclockwise, and with appropriate cancellation of positive and negative area being represented visually. It'd be especially nice if the demo handled reentrant polygons sensitively, so that regions encircled m times had a different color than regions encircled n times, when the integers m and n (positive, negative, or zero) are unequal. Jim Propp

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[math-fun] ChipChip
by James Propp 24 Jul '16

24 Jul '16

I have an idea for an app that a friend of mine has implemented in CSS: http://mathenchant.org/chipchip/ I’d be interested in people’s comments. Is it fun to play? Are the directions clear? I’m aware of glitches in the implementation (some of which are browser-specific or platform-specific), and do let me know about them, but those are of secondary interest; what I really want is ideas for making the app more fun, and (from those of you with some relevant experience) a sense of whether the game has commercial potential. The page runs on laptops and smartphones. If the boundaries between tiles aren't clear, try a different browser (and let me know about the issue). ChipChip started out as a math game, but James Tanton pointed out that there’s a much bigger market for anagram-syle word-games. So you can play in either Number Mode (the goal is to sort the numbered tiles into ascending order) or Word Mode (the goal is to spell a word). I specifically chose words with no repeated letters; do you think it would be more fun if I dropped this constraint? Jim Propp

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[math-fun] permanently excited tantalum
by Bill Gosper 21 Jul '16

21 Jul '16

"The most long-lived non-ground state <https://en.wikipedia.org/wiki/Ground_state> nuclear isomer is the nuclide tantalum-180m <https://en.wikipedia.org/wiki/Tantalum-180m>(180m 73Ta <https://en.wikipedia.org/wiki/Tantalum-180m>), which has a half-life <https://en.wikipedia.org/wiki/Half-life> in excess of 1,000 trillion years. This nuclide occurs primordially, and has never been observed to decay to the ground state. (In contrast, the ground state nuclide tantalum-180 does not occur primordially, since it decays with a half life of only 8 hours to 180Hf (86%) or 180W (14%))." More: https://en.wikipedia.org/wiki/Isotopes_of_tantalum So where do you get the radioactive stuff? And how do you rescue (excite) it? --rwg

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[math-fun] (no subject)
by Michael Kleber 21 Jul '16

21 Jul '16

Nice Nature article on the study of bicycle dynamics: http://www.nature.com/news/the-bicycle-problem-that-nearly-broke-mathematic… Excerpt: """ He first rewrote the bicycle equations in terms of the caster trail, the crucial variable that Jones had championed. He expected to find that if the trail was negative, the bicycle would be unstable, but his calculations suggested otherwise. In a report that he prepared at the time, he sketched a bizarre bicycle with a weight jutting out in front of the handlebars. “A sufficiently forward [centre of mass] can compensate for a slightly negative trail,” he wrote. No single variable, it seemed, could account for self-stability. This discovery meant that there was no simple rule-of-thumb that could guarantee that a bike is easy to ride. Trail could be useful. Gyroscopic effects could be useful. Centre of mass could be useful. For Papadopoulos, this was revelatory. The earliest frame builders had simply stumbled on a design that felt OK, and had been riding around in circles in that nook of the bicycle universe. There were untested geometries out there that could transform bike design. """ --Michael -- Forewarned is worth an octopus in the bush.

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[math-fun] Numbers of the form K^2 + KL + L^2
by Dan Asimov 20 Jul '16

20 Jul '16

Numbers of the form K^2 + KL +L^2, K and L integers, arise often when making calculations connected with the triangular lattice in the plane. Let tau = exp(2pi*i/6). For K, L > 0, K^2 + KL +L^2 is, among other things, the number of points in the quotient ring Z{tau] / <K + L*tau> namely, the Eisenstein integers factored out by its ideal <K + L*tau>. I would like to know more about these numbers. For example: * Given any integer N, is there a simple test for whether N is of the form K^2 + KL + L^2 for some integers K, L ??? * What can be said about the prime factorization of K^2 + KL + L^2 ??? * Is there a relationship of such numbers to Eisenstein primes? —Dan

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Re: [math-fun] Wolfram Alpha stumper
by Bill Gosper 20 Jul '16

20 Jul '16

On 2016-07-14 15:24, David Wilson wrote: > solve x^y = x + y = 2 > Can you blame it? gosper.org/x^y=x+y=2.png (reciprocated) is a pretty hairy root system. LambertW seems to fall short. But what if we could solve Kepler's M = E - ε sin E? (Which I expected to arise in our discussion of the average position of the Earth.) --rwg

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[math-fun] Proof or idea for proof needed
by Zak Seidov 20 Jul '16

20 Jul '16

Just got from unknown guy. Any ideas? -------- Пересылаемое сообщение -------- От кого: grafix(a)csl.pl Кому: zakseidov(a)yahoo.com Дата: Среда, 20 июля 2016, 0:08 +03:00 Тема: Proof or idea for prrof needed Dear Zak, Do You know how to prove that z(m,n) for integer m,n defined as z=(n (2 m + 11 n)^5)/(64 (m^2 + 261 m n - 2501 n^2)^2 (m^2 + n^2)) can't be integer with trivial exception {z=0, m=-11,n=2} or {z=0, m=11,n=-2} Best wishes Artur Jasinski, Poland ---------------------------------------------------------------------- ----------------------------------------------------------------------

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[math-fun] A sequence concerning the abundancy index (long)
by Allan Wechsler 20 Jul '16

20 Jul '16

I apologize for the blurriness of the following prose. There are no difficult concepts here, but I haven't been able to express it in English very well. Read carefully; I assure you there is a reward here. This came from my amateur fossickings in the world of multiply-perfect numbers. For n a positive integer, let S(n) be the sum of the divisors of n; I'm using S because I can't easily type a sigma. The abundancy index of n, T(n), is the rational number S(n)/n. Classical perfect numbers have T(n) = 2; the more general multiply-perfect numbers have integer T(n). All of this so far is standard. Suppose T(n) = U(n)/V(n), with the fraction expressed in lowest terms. Factor U(n) and V(n) into prime powers. Of these prime powers, one of them is the largest. Sometimes it is from the numerator, perhaps from the denominator. (The sequence of n where the largest prime power in T(n) is found the denominator, in this way, is 5, 8, 11, 13, 14, 17, 19 ..., which is not in OEIS, but that's not why I called you all here today.) Suppose the largest prime power of T(n) is p^e. Let W(n) be defined as np if p^e was found in the numerator of T(n), and n/p if it was found in the denominator. An example: Let n = 168 = 2^3.3.7; then S(n) = 480 = 2^5.3.5; and T(n) = 480/168 = 12/7 = 2^2.3.7^(-1). Of the three component prime powers, 3, 4, and 7, 7 is the largest and it is found in the denominator. So n' = n/7 = 24. The sequence I want to consider is the trajectory of 2 under iterated applications of W. It starts 2, 6, 12, 84, 168, 24 ... and of course is not in OEIS. Each term is obtained from the previous term by multiplying or dividing by a prime; to reiterate, the prime is the base of the largest prime power in T(n), and we multiply or divide according to which side of the fraction line we find it on. So let A(1) = 2, A(i) = W(A(i-1)). The sequence flops around randomly, growing and shrinking but mostly growing, and becoming more and more exuberantly composite. A(75) picks up the prime factor 2147483647 = 2^31 - 1. The sum of divisors of this is, of course, 2^31, and other multiply-even factors of A(75) drive the power of 2 up to 2^47. At this point the trajectory enters a long episode dominated by this large power of 2 in the numerator, doubling at each step. I was sure that before this reservoir of 2's was exhausted, we would pick up another multiply-even factor and the sequence would climb forever. I was mistaken. The glut of 2's ends at A(93); the "bull market" gives out and there is a crash in which we lose, first that big factor of 2147483647, and then lots of 2's and other detritus that we've been accumulating along the way. I'm up to A(116), which, in case you care, is 2^39.3^5.5^2.7^2.11.13^2.19^2.31^2.61.83.127.331.9719 . The exponent of 2 peaked at 47 and has been declining fitfully ever since. If it ever gets up to 60 it will acquire another Mersenne factor, sparking another bubble, but the exponent may bop around between 30 and 60 forever (with the sequence eventually entering a cycle). I don't know if I have the patience to find out. But the long spells of growth and decay are certainly intriguing. I was led to this sequence because one way to find multiply perfect numbers is to start with a candidate n, and repeatedly adjust the exponents of its prime factors in a way suggested by the non-integralness of T(n), the goal being to whittle T(n) down to an integer. One naturally tries to automate the process of selecting which prime to adjust, which leads to sequences like A(i).

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